Transcribed Text
10. Prove the following generalization of Proposition 2.20. Let G and 2 be
open in C and suppose f and h are functions defined on G, g : 2 C and
suppose that f (G) C S. Suppose that g and h are analytic, 8' (w) + 0 for any
w, that f is continuous, h is oneone, and that they satisfy h(z) = 8(f(2)) for
Z in G. Show that f is analytic. Give a formula for f'(z).
11. Suppose that f: G
C is a branch of the logarithm and that n is an
integer. Prove that z" = exp (nf(z)) for all Z in G.
12. Show that the real part of the function 2/2 is always positive.
13. Let G = C  { Z € R : Z V 0} and let n be a positive integer. Find all
analytic functions f: G
C such that Z = (f(z))" for all Z € G.
14. Suppose f: G
C is analytic and that G is connected. Show that if
f (z) is real for all Z in G then f is constant.
{
1
15. For r > 0 let A =
w : w = exp

where 0 < 1/11 <
r
; determine the
Z
set A.
G maximal? Are f and g analytic?
17. Give the principal branch of V 1  z.
19. Let G be a region and define G* = If f: G C is analytic
prove that f * : G*
C, defined by f* *(z) = f(z), is also analytic.
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